Where do all these formulas really come from?

So how did people like Einstein, and Newton really do it on a day to day basis? How do physicists really do physics? Is it a bunch of scientifically minded mathematicians sitting around doing lots of math until the results they get match experimental ones, or is it more a matter of doing things like plotting data on graphs, coming up with best fit lines and then finding the formula for the line like we all do in high school math class?

I suppose a strong interest and aptitude in mathematics is most important, but I also see as vital a strong desire to find and define mathematical relationships.
Perhaps those people see a repeatable phenomenon, find out that that it has not previously been described mathematically, and then go for it... days, months even many years. Sometimes with collaboration.

So how did people like Einstein, and Newton really do it on a day to day basis? How do physicists really do physics? Is it a bunch of scientifically minded mathematicians sitting around doing lots of math until the results they get match experimental ones, or is it more a matter of doing things like plotting data on graphs, coming up with best fit lines and then finding the formula for the line like we all do in high school math class?

Well, no. Curve-fitting would be pure empiricism. While that's sometimes useful, you can only find really obvious relationships with it. Otherwise you're just stuck with a re-statement of what you already knew: "The curve looks like this". Empirical formulas can sometimes help inspire theories (e.g. Rydberg's formula inspired the Bohr model of the atom), but they're not scientific theories themselves. They're just a sort of summation of observed information.

Physical theories start with a theory - an idea about how things work. Basically "What if...". This idea can then be put into mathematical form, and using maths/logic you work out the consequences of this assumption and get some predictions. If they match the experimental results, or at least tell us something about the experimental results, then it's a useful theory. If not, then it gets thrown in the wastepaper basket. (It's easy to get the idea from textbooks that Science follows a nice straight path. In reality it's a very twisted path, full of dead-ends and failed attempts. It's just that those get forgotten.)

Here is one specific example. In 1696, Isaac Newton was challenged to solve the following problem. Consider a bead sliding without friction on a curved wire under the influence of gravity from some starting point x,y to a lower point x', y'. Find the shape (path) of the wire such that the transit time of the bead is a minimum.

Newton thought about it, and (supposedly) solved it in a day. (How long would it take you?) The problem is now known as the Brachistochrone problem. See

A theory is still just a theory until it gets compared to data points. At some point, you need curve fitting or some other kind of comparison with experimental data. Whether it's empirical or not depends on how the fitting function is defined. If it is defined only by the fit, it's empirical. But if it can be calculated from an underlying theory, and the resulting curve fits the data points, then we're getting serious. Additional fitting curves from the same theory will serve to validate it. This is physics.

The game is often to construct an equation that, when when solved with the proper boundary conditions, gives functions that can be fitted with experimental data.

Of the top of my head, this applies to at least to Newton's laws, Maxwell's equation's, and Shrodinger's equation. Einstein's GR would also be useless if hadn't at some point been compared to experimental data (astronomical observation in this case).

But IMO, finding the right equations is simply a matter of educated guess (or well-thought out) work by people who are familiar with the mathematical tools.

"Physical theories start with a theory - an idea about how things work. Basically "What if...". This idea can then be put into mathematical form, and using maths/logic you work out the consequences of this assumption and get some predictions."

This is the part I'd like to examine. Turning an idea into a mathematical equation. At first, it seems pretty straight forward when you're dealing with simple and easily observable phenomena: "If I push twice as hard, the object will move twice as fast." But the universe rarely seems so simply, and yet it seems that many physics formulas boil down to simple stuff like inverse square laws and straight direct proportionality. I mean most of Newtonian kinematics is simple multiplication and division, half of this times the square of that, and so on...

Staff: Mentor

A theory is still just a theory until it gets compared to data points. At some point, you need curve fitting or some other kind of comparison with experimental data. Whether it's empirical or not depends on how the fitting function is defined. If it is defined only by the fit, it's empirical. But if it can be calculated from an underlying theory, and the resulting curve fits the data points, then we're getting serious. Additional fitting curves from the same theory will serve to validate it. This is physics.

The game is often to construct an equation that, when when solved with the proper boundary conditions, gives functions that can be fitted with experimental data.

That's not what "curve fitting" is. Curve fitting is a mathematical method for building an equation to fit data. If you already have the equation and are trying to check it, you simply plug-and-chug and compare the predicted result to the experimental one.

"Physical theories start with a theory - an idea about how things work. Basically "What if...". This idea can then be put into mathematical form, and using maths/logic you work out the consequences of this assumption and get some predictions."

Agreed

This is the part I'd like to examine. Turning an idea into a mathematical equation. At first, it seems pretty straight forward when you're dealing with simple and easily observable phenomena: "If I push twice as hard, the object will move twice as fast." But the universe rarely seems so simply, and yet it seems that many physics formulas boil down to simple stuff like inverse square laws and straight direct proportionality. I mean most of Newtonian kinematics is simple multiplication and division, half of this times the square of that, and so on...

When I first came across Newton's solution to the Brachisochrone problem (post #4), I could not believe that math could be so powerful. It certainly involves a lot more than simple multiplication and division. Newton had to "invent" calculus of variations in order to solve it. And he did it in (rumored to be one day in) 1696.

That's not what "curve fitting" is. Curve fitting is a mathematical method for building an equation to fit data. If you already have the equation and are trying to check it, you simply plug-and-chug and compare the predicted result to the experimental one.

If you have a theoretical function that fits the first time, then great. If not, curve fitting may help make corrections to the model. Either way, the goal is to find the function that fits the data, and to be able to derive that function from general postulates. How to do that is up to you.

First you guess. Don't laugh, this is the most important step.
Then you compute the consequences. Compare the consequences to
experience. If it disagrees with experience, the guess is wrong.
In that simple statement is the key to science. It doesn't matter
how beautiful your guess is or how smart you are or what your name is.
If it disagrees with experience, it's wrong. That's all there is to it.